The inequality −2(x + 10) ≥ 75 says the same thing as −2x − 20 ≥ 75. I can multiply by -2 on the left side without reversing the inequality symbol.
Yes, that is true. But when you divide both sides by (-2) to solve for x; you will have to reverse inequality sign.
-2x - 20 >= 75.
-2x >= 95
X <= -47.5.
Therefore, all values of x that are less than or equal to (-47.5) will satisfy the given inequality.
Well, multiplying by a negative number does have a special relationship with inequalities. When you multiply both sides of an inequality by a negative number, you actually need to reverse the inequality symbol. So, in this case, if you multiply both sides by -2, you should actually reverse the inequality symbol to maintain the same meaning. Therefore, −2(x + 10) ≥ 75 is not the same thing as −2x − 20 ≥ 75. It's like trying to reverse the direction of a one-way street - it just doesn't work out so well.
To determine whether the inequality −2(x + 10) ≥ 75 is equivalent to −2x − 20 ≥ 75, let's simplify both inequalities step-by-step:
1. Original inequality: −2(x + 10) ≥ 75
Start by distributing the -2 to both terms inside the parentheses:
−2x - 20 ≥ 75
2. Now we have −2x - 20 ≥ 75.
To determine if you can multiply by -2 on the left side without reversing the inequality symbol, we need to consider a general rule:
If you multiply both sides of an inequality by a positive number, the inequality symbol remains the same. However, if you multiply both sides by a negative number, the inequality symbol must be reversed.
Since -2 is negative, you must reverse the inequality symbol when multiplying.
3. Multiply both sides of the inequality by -1 (-2/-2 = 1):
(-2) * (-2x - 20) ≤ (-2) * (75)
Simplifying the inequality gives:
2x + 20 ≤ -150
4. Remove the constant 20 by subtracting it from both sides:
2x + 20 - 20 ≤ -150 - 20
This simplifies to:
2x ≤ -170
5. Finally, divide both sides of the inequality by 2 to isolate x:
(2x)/2 ≤ (-170)/2
The inequality becomes:
x ≤ -85
So, after performing all the steps, the inequality -2(x + 10) ≥ 75 is not equivalent to -2x - 20 ≥ 75. Instead, it simplifies to x ≤ -85.
In order to understand if the inequality −2(x + 10) ≥ 75 is equivalent to −2x − 20 ≥ 75, let's break it down step by step.
1. Starting with the first inequality: −2(x + 10) ≥ 75.
2. Apply the distributive property by multiplying −2 to both terms inside the parentheses:
−2 * x + 2 * (-10) ≥ 75.
Simplify this to: −2x - 20 ≥ 75.
3. Now, let's compare the second inequality, −2x - 20 ≥ 75, with the original inequality.
4. The expression obtained through simplification in step 2 is indeed equivalent to −2x - 20 ≥ 75.
So, you are correct.
Regarding reversing the inequality symbol when multiplying both sides by a negative number, it is important to note that reversing the inequality symbol only occurs when multiplying or dividing both sides of an inequality by a negative number. In this case, you are multiplying both sides by -2, which is a negative number. However, you are multiplying the entire left side of the inequality. Since the entire left side is being multiplied by the same negative number, there is no need to reverse the inequality symbol.
So, your understanding is correct, and multiplying by -2 on the left side without reversing the inequality symbol is valid in this case.